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80. To find the Angle whose Logarithmic Cosine is Given. Find the angle whose log cosine is 9.8934342. We have from

page

113

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Let d=diff. between 38° 31' and required angle; then

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Note. — In using both the tables of the natural sines, cosines, etc., and the tables of the logarithmic sines, cosines, etc., the student will remember that, in the first quadrant, as the angle increases, the sine, tangent, and secant increase, but the cosine, cotangent, and cosecant decrease.

EXAMPLES.

1. Given log sin 14° 24'= 9.3956581,

log sin 14° 25'= 9.3961499; find the angle whose log sine is 9.3959449. Ans. 14° 24' 35".

2. Given log sin 71° 40'= 9.9773772,

log sin 71° 41'=9.9774191; find the angle whose log sine is 9.9773897.

71° 40' 18".

3. Given log cos 28° 17'= 9.9447862,

log cos 28° 16'= 9.9448541; find the angle whose log cosine is 9.9448230.

28° 16' 27".5.

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4. Given log cos 80° 53'= 9.1998793,

log cos 80° 52' 50"= 9.2000105; find the angle whose log cosine is 9.2000000.

Ans. 80° 52' 51".

5. Given log tan 35° 4'=9.8463018,

log tan 35° 5'= 9.8465705; find the angle whose log tangent is 9.8464028.

35° 4' 23".

6. Given log sin 44° 35' 30"=9.8463678,

log sin 44° 35' 20"= 9.8463464; find the angle whose log sine is 9.8463586.

44° 35' 25".7.

7. Find the angle by page 113 whose log tangent is 10.1018542.

Ans. 51° 39' 28''.7.

81. Angles near the Limits of the Quadrant. — It was assumed in Arts. 72–80 that, in general, the differences of the trigonometric functions, both natural and logarithmic, are approximately proportional to the differences of their corresponding angles, with certain exceptions. The exceptional cases are as follows:

(1) Natural functions. For the sine the differences are insensible for angles near 90°; for the cosine they are insensible for angles near 0°. For the tangent the differences are irregular for angles near 90°; for the cotangent they are irregular for angles near 0°.

(2) Logarithmic functions. — The principle of proportional parts fails both for angles near 0° and angles near 90°. For the log sine and the log cosecant the differences are irregular for angles near 0°, and insensible for angles near 90°. For the log cosine and the log secant the differences are insensible for angles near 0°, and irregular for angles near 90°. For the log tangent and the log cotangent the differences are irregular for angles near 0° and angles near 90°.

It follows, therefore, that angles near 0° and angles near 90° cannot be found with exactness from their log trigonometric functions. These difficulties may be met in three ways.

(1) For an angle near 0° use the principle that the sines and tangents of small angles are approximately proportional to the angles themselves. (See Art. 130.)

(2) For an angle near 90o use the half angle (Art. 99).

(3) In using the proportional parts, find two, three, or more orders of differences (Alg., Art. 197).

Special tables are employed for angles near the limits of the quadrant.

EXAMPLES.

1. Given log10 7=.8450980, find log10 343, log 10 2401, and log10 16.807.

Ans. 2.5352940, 3.3803920, 1.2254900.

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2. Find the logarithms to the base 3 of 9, 81, 1, 2, .1, 1:

Ans. 2, 4, -1, -3, -2, - 4. 3. Find the value of log, 8, logo.5, log: 243, log: (.04), logo 1000, log10.001.

Ans. 3, - 1, 5, -2, 3, – 3.

. 4. Find the value of log, a), log, Vo’, log: 2, logz 3, log10 10.

Ans. , , }, }, 1:

1 Given logio2 =.3010300, log103= .4771213, and logio 7 = .8450980, find the values of the following:

4

2

1

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5. log10 35, logio 150, logio.2.

Ans. 1.544068, 2.1760913, 1.30103.

6. log103.5, log10 7.29, log10.081.

Ans. .5440680, .8627278, 2.9084852. 7. log10 }, log10 35, log10 12.

Ans. .3679767, 2.3856065, .0780278.

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8. Write down the integral part of the common logarithms of 7963, .1, 2.61, 79.6341, 1.0006, .00000079.

Ans. 3, -1, 0, 1, 0, -7. 9. Give the position of the first significant figure in the numbers whose logarithms are

2.4612310, 1.2793400, 6.1763241. 10. Give the position of the first significant figure in the numbers whose logarithms are 4.2990713, .3040595, 2.5860244, 3.1760913, 1.3180633, .4980347. Ans. ten thousands, units, hundreds, 3rd dec. pl., 1st

dec. pl., units. 11. Given log 7 =.8450980, find the number of digits in the integral part of 789, 499, 343*, (49)", (4.9)", (3.43) "0.

Ans. 9, 11, 85, 4, 9, 6. 12. Find the position of the first significant figure in the numerical value of 20", (.02), (.007)”, (3.43) 16, (.0343),

100

20

12

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(.0343) Io.

Ans. tenth integral pl., 12th dec. pl., 5th dec. pl., units,

12th dec. pl., 1st dec. pl.

Show how to transform

13. Common logarithms to logarithms with base 2.

Ans. Divide each logarithm by .30103. 14. Logarithms with base 3 to common logarithms.

Ans. Multiply each log by .4771213. 15. Given log10 2 =.3010300, find log, 10. 3.32190. 16. Given log10 7 = .8450980, find log, 10.

1.183. 17. Given log10 2 = .3010300, find logg 10. 1.10730.

18. The mantissa of the log of 85762 is 9332949; find (1) the log of ".0085762, and (2) the number of figures in (85762)", when it is multiplied out.

Ans. (1) 1.8121177, (2) 55. 19. What are the characteristics of the logarithms of 3742 to the bases 3, 6, 10, and 12 respectively?

Ans. 7, 4, 3, 3. 20. Prove that 7 log 16 + 6 log | + 5 log + log 3=log 3.

1 21. Given log10 7, find log, 490.

Ans. 2 +

log10 7 22. From 5.3429 take 3.6284.

3.7145.

2.4076.

23. Divide 13.2615 by 8.
24. Prove that 6 log + 4 log % + 2 log 2 = 0.
25. Find log {297 V11;$ to the base 3V11.

1.8.

Given log 2 =.3010300, log 3 = .4771213.

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26. Find log 216, 6480, 5400, g.

Ans. 2.3344539, 3.8115752, 3.7323939, 1.6478174. 27. Find log.03, 6-4, (53)-1.

Ans. 2.4771213, 1.7406162, 1.6365006. 28. Find log .18, log 2.4, log 36

Ans. 1.2552726, .3802113, 1.2730013. 29. Find log (6.25)}, log 4V.005. .1136971, 1.45154. 30. Given log 56321 = 4.7506704,

log 56322 = 4.7506781; find log 5632147.

6.7506740. 31. Given log 53403 = 4.7275657,

log 53402 = 4.7275575; find log 5340234.

6.7275603. 32. Given log 56412 = 4.7513715,

log 56413=4.7513792; find log 564.123.

2.7513738.

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